Selection combining for noncoherent decode and forward relay networks

R E S E A R C H

Open Access

Selection combining for noncoherent

decode-and-forward relay networks

Ha X Nguyen

and Ha H Nguyen

Abstract

This paper studies a new decode-and-forward relaying scheme for a cooperative wireless network composed of one source,Krelays, and one destination and with binary frequency-shift keying modulation. A single threshold is employed to select retransmitting relays as follows: a relay retransmits to the destination if its decision variable is larger than the threshold; otherwise, it remains silent. The destination then performs selection combining for the detection of transmitted information. The average end-to-end bit-error-rate (BER) is analytically determined in a closed-form expression. Based on the derived BER, the problem of choosing an optimal threshold or jointly optimal threshold and power allocation to minimize the end-to-end BER is also investigated. Both analytical and simulation results reveal that the obtained optimal threshold scheme or jointly optimal threshold and power-allocation scheme can significantly improve the BER performance compared to a previously proposed scheme.

Keywords:cooperative diversity, frequency-shift keying, fading channel, decode-and-forward protocol, selection combining, power allocation

1 Introduction

Cooperative diversity has recently emerged as a promis-ing technique to combat fadpromis-ing experienced in wireless transmissions. The basic idea behind this technique is that a source node cooperates with other nodes (or relays) in the network to form a virtual multiple antenna system [1-7], hence providing spatial diversity. Amplify-and-forward (AF) and decode-Amplify-and-forward (DF) are two well-known protocols to realize cooperative diversity. In AF, the relays amplify and forward the received signals to the destination. In DF, the received signal at each relay node is first decoded, and then remodulated and retransmitted. Unlike the AF protocol, it is not simple to provide cooperative diversity with the DF protocol. This is due to possible retransmission of erroneously decoded bits of the message by the relays in the DF pro-tocol [1,4,8,9].

On the other hand, the issue of how to efficiently combine multiple received signals at the receiver is of practical interest and has been intensively studied, both in point-to-point and relay communication systems. Typical combining schemes include maximal ratio

combining (MRC), equal gain combining (EGC), and selection combining (SC). Since SC processes only one of the received signals, it is the simplest when compared to other combining schemes [10]. In fact, SC scheme has been widely investigated for coherent DF coopera-tive systems in which a perfect knowledge of channel state information (CSI) is available at the receivers (at relays and destination) [11-14]. Moreover, the SC tech-nique is especially suitable in noncoherent communica-tions because instead of selecting the largest signal-to-noise ratio as in coherent systems, the signal branch with the largest signal-plus-noise power can be selected. Due to these advantages, the SC scheme for binary non-coherent frequency-shift keying (FSK) in point-to-point communications has also been well studied in the litera-ture [15-18].

The majority of research works in wireless relay net-works is for coherent communications. Since obtaining the channel state information (CSI) in coherent commu-nications might be unrealistic in fast fading environment and in multiple-relay networks, there have been some recent works that exploit noncoherent modulation and demodulation in cooperative networks [19-25]. In what follows, related works and the contributions of this paper are described.

* Correspondence: [email protected]

Department of Electrical and Computer Engineering, University of Saskatchewan 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada

1.1 Related works

Differential phase-shift keying (DPSK) has been studied for both AF and DF protocols in [19-22]. However, with the DF protocol in [20], the authors considered an ideal case that the relay is able to know exactly whether each decoded symbol is correct. The works in [21,22] exam-ine a very simple cooperative system with one source, one relay, and one destination node. Optimal resource allocation has been studied for noncoherent systems in [23,24] to further improve the error performance of the system when DPSK is employed.

A framework of noncoherent cooperative relaying for the DF protocol employing FSK has been studied in [25] in which the maximum likelihood (ML) demodulation was developed to detect the signals at the destination. Due to the nonlinearity form and high complexity of the ML scheme, a suboptimal piecewise-linear (PL) scheme was also proposed in [25] and shown to perform very close to the ML scheme. It is noted, however, that a closed-form BER approximation for either the ML or PL scheme in [25] is not readily available for networks with more than two relays. Furthermore, the BER perfor-mance with either ML or PL demodulation can still suf-fer from the error propagation phenomenon [6].

To address the issue of error propagation and inspired by the work in [6], reference [26] examines an adaptive noncoherent relaying scheme in which two thresholds are employed at the relays and destination as follows. One threshold is used to select retransmitting relays: a relay retransmits to the destination if its decision vari-able is larger than the threshold, otherwise it remains silent. The other threshold is used at the destination for detection: the destination marks a relay as a retransmit-ting relay if the decision variable corresponding to the relay is larger than the threshold, otherwise, the destina-tion marks it as a silent relay. Then, the destinadestina-tion sim-ply combines (in a ML sense) the signals from the retransmitting relays and the signal from the source to make the final decision. Numerical results in [26] show that, with optimal threshold values, the cooperative relaying scheme proposed in [26] can significantly improve the error performance over the schemes in [25]. Unfortunately, closed-form BER expressions are only available for the single-relay and two-relay net-works in [26]. As such, the important task of optimizing the threshold values has to rely on numerical search for networks with more than two relays.

1.2 Contributions

This paper is also concerned with a threshold-based relaying scheme for noncoherent DF cooperative net-works in which binary FSK (BFSK) is employed at the source and relays. The transmission protocol considered is as follows. After receiving the signal from the source

in the first phase, each relay decides to retransmit the decoded information if its decision variable is higher than a threshold. Otherwise, it remains silent in the sec-ond phase. At the destination, selection combining is

employed to select the “strongest” received signal to

decode.

It should be pointed out that one practical aspect of the proposed scheme is that the destination has no information on whether a particular relay retransmits or remains silent in the second phase. This means that the destination does not known whether a received signal is from a retransmitting relay or a silent relay. Therefore, the destination might select a signal from the relay that remains silent to decode. However, this possibility hap-pens with a very small probability due to the selection rule implemented at the destination.

The main difference between the protocol in this paper and the one in [26] is that no threshold is needed and selection combining is performed at the destination. This simpler protocol (as compared to the protocol in [26]) also allows one to obtain a closed-form BER

expression for a general network with K relays. This

leads to a convenient optimization of threshold and

power allocation among K relays. Numerical results

show that our BER expression is accurate. Moreover, our proposed protocol provides a superior performance under all channel conditions with similar complexity compared to the piecewise-linear (PL) receiver in [25].

1.3 Organization of the paper

The remainder of this paper is organized as follows. Sec-tion 2 describes the system model. SecSec-tion 3 presents the BER computation and discusses how to find the optimal threshold and power allocation. Numerical and simulation results are presented in Section 4. Finally, Section 5 concludes the paper.

2 System model

Consider a wireless communication system in which the source node sends its message to the destination node

through K relay nodes. All nodes operate in a

half-duplex mode, i.e., a node cannot transmit and receive simultaneously and DF protocol is employed at the relays. We consider that the relays retransmit signals to

the destination in orthogonal channelsaand there is no

direct link between the source and destination. For con-venience, the source, relays, and destination are denoted

and indexed by node 0, node i, i = 1, ..., K, and node

K+ 1, respectively.

Signal transmission from the source to destination is completed in two phases as illustrated in Figure 1. In the first phase, the source broadcasts a BFSK signal and

the baseband received signals at nodei,i= 1, ..., K, are

y0,i,0= (1−x0)

E0h0,i+n0,i,0, (1)

y0,i,1=x0

E0h0,i+n0,i,1, (2)

where h0,i andn0,i,k denote the channel fading

coeffi-cient between node 0 and node iand the noise

compo-nent at nodei, respectively.E0is the average transmitted

symbol energy of the source. The third subscriptkÎ {0,

1} in (1) and (2) denotes the two frequency subbands used in BFSK signaling. Furthermore, the source symbol

x0= 0 if the first frequency subband is used andx0= 1 if

the second frequency subband is used.

With noncoherent BFSK, signal detection at the ith

relay node is carried out by simply comparing the signal energies received in the two subbands. As such the instantaneous magnitude of the energy difference in the

two subbands, namelyθ0,i= ||y0,i,0|2- |y0,i,1|2|, serves as

a reliability measure of the detection at the ith relay.

Similar to [26], node ionly decodes and retransmits a

BFSK signal ifθ0,i> θrth, whereθrthis some fixed

thresh-old value to be determined. If nodei transmits in the

second phase, the received signals at the destination in the two subbands are given by

yi,K+1,0= (1−xi)

Eihi,K+1+ni,K+1,0, (3)

yi,K+1,1=xi

Eihi,K+1+ni,K+1,1, (4)

where Ei is the average transmitted symbol energy

sent by nodeiand ni,K+1,kis the noise component at the

destination in the second phase. Note that if the ith

relay makes a correct detection, thenxi=x0. Otherwise

xi ≠ x0. On the other hand, when θ0,i< θrth, node i

remains silent. In this case, the outputs in the two sub-bands are given by

yi,K+1,0=ni,K+1,0, (5)

yi,K+1,1=ni,K+1,1. (6)

After receiving all the signals from the relays, the des-tination chooses only one signal with the largest magni-tude of the energy difference in the two subbands to

decode. In other words, the signal from nodeiis chosen

if maxj≠iθj,K+1 <θi,K+1 where θj,K+1 = ||yj,K+1,0|2 - |yj,K

+1,1|2|,j= 1, ...,K. The detector is of the form:

= |yi,K+1,0|2− |yi,K+1,1|2 0 ≷ 1

0. (7)

The next section derives the average BER for a general network, i.e., a network with arbitrary qualities of source-relay and relay-destination links. Using the derived BER, the optimum thresholds can then be numerically found.

3 BER analysis and optimization of threshold and power allocation

Let the noise components at the relays and destination

be modeled asb i.i.d.CN(0,N0)random variables. The

channel between any two nodes is Rayleigh flat fading,

modeled asCN(0,σ2

i,j), wherei,j refer to transmit and

receive nodes, respectively. The instantaneous received

SNR for the transmission from nodeito nodejis given

as gi,j=Ei|hi,j|2/N0 and the corresponding average SNR

isγ¯i,j=Eiσ_i2_,j/N0. With Rayleigh fading, the pdf ofgi,jis

fi,j(γi,j) = _γ¯1_i,je−γi,j/γ¯i,j.

Source

th

0,1 r

θ >θ

0,K

y

Decode and Re-transmit

Discard N

Y

th

0,K r

θ >θ Decode and _Re-transmit

Discard N

Y Relay 1

Relay K

Selection Combining

Detection Destination

Recall that the destination selects only one signal

among Kreceived signals to decode. The selected relay

might forward a correct bit, an incorrect bit or remain silent in the second phase. Therefore, there are three different cases that result in different BERs at the

desti-nation. We parameterize the three cases byΘÎ{1, 2, 3}

whereΘ= 1,Θ= 2, and Θ= 3 are the events that the

selected relay forwards a correct bit, an incorrect bit and remains silent, respectively. By using the law of total probability, the average BER with a given threshold

θth

r can be expressed as

BER (θ_rth) =

To compute all the terms in (8), divide the setSrelay=

{1, 2, ...,K} of Krelays into three disjoint subsetsΩ1,Ω2,

andΩ3, which include the relays that forward a correct

bit, an incorrect bit, and remain silent in the second

phase, respectively. Clearly,K=|Ω1|+|Ω2|+|Ω3|where

|Ω|denotes the cardinality of setΩ. Without loss of

gen-erality, assume that the transmitted information bit is“0”.

Also let Wm (m Î Ω1), Vn (n Î Ω2) and Rl (l Î Ω3)

denote the energy differences in the two subbands measured at the destination for relay-destination links

involving the relays in setsΩ1, Ω2andΩ3, respectively.

ObviouslyP(ε,Θ=i) can be calculated as follows:

the conditional BER and case probability for the specific

set (Ω1, Ω2, Ω3). The notation P(A)means the power

set of its argument, i.e., the set of all its subsets

(includ-ing the empty set∅).A\B denotes the relative

comple-ment of the setBin the setA.

First, according to Lemmas 2 and 4 in [26], the

prob-ability density functions (pdfs) of Wm, Vn and Rl are

It then follows that

f_|Wm|(x) =

The probability of occurrence for the specific set {Ω1,

Ω2,Ω3} can be determined to be

tude of the energy difference in the two subbands at

nodeiis smaller than the threshold, i.e.,θ0,i< θrth. The

the energy difference in the two subbands is larger than

the threshold, i.e., θ0,i> θrth. Therefore, I2(θrth,γ¯0,i)can

Next we compute the average BER for Θ = 1

condi-tioned on {Ω1, Ω2, Ω3}. In this case, the selected relay

forwards a correct bit. This means that an error occurs

at the destination if among the KstatisticsWm, Vnand

Rl, the one with the largest magnitude is one of

Wm and negative. Thus, the conditional BER can be

where Wm= maxi=m(|Wi|,|Vn|,|Rl|). The pdf of Wm

can be found as follows:

fWm(x) =

It then follows that

P1,2,3(ε,= 1) =

are three disjoint subsets of P()and the union of

those disjoint subsets isΩ.

3.3 CaseΩ= 2

In this case, the selected relay forwards an incorrect bit,

i.e., an error occurs if among the Kstatistics Wm, Vn

selected relay remains silent in the second phase, i.e., it

is one of the relays inΩ3. The conditional BER is

P1,2,3(ε,= 3) =

be found as follows:

fRl(x) =

To summarize, all the expressions involved in the final expression of the average BER in (8) can be calculated analytically. Although final expression is quite involved and presents limited insights, it is simple enough to use

in optimizing the threshold_θth

r to minimize the average

BER of the network.

First, for a fixed power allocation among the source and relays, the optimization of the threshold value can be set up as follows:

ˆ

On the other hand, the total transmitted power of the network can also be optimally allocated to the source and relays. To this end, let the total signal energies at

the source and relays beEtotal and the maximum signal

energy that can be allocated to node ias Ei, max. Then,

the joint optimization of the threshold_θth

r and power to

minimize the average BER are as follows:

With the closed-form expression of the average BER, the above optimization problems can be solved by opti-mization techniques such as the Lagrange method [27]. Unfortunately, the exponential terms in the final expres-sions render a closed-from solution intractable. The optimization problems in (26) and (27) are simply

solved with the MATLAB Optimization Toolbox.c It

should be pointed out that, without proving the BER function is convex, the solutions obtained by MATLAB might only locally optimum solutions. Nevertheless, plotting the BER function versus the threshold value for various power allocations shows that the objective func-tion is convex. This strongly suggests that the solufunc-tions are globally optimum. Moreover, since the average BER formulated in (8) only requires information on the aver-age SNRs of the source-relay and relay-destination links, the optimization problems can be solved off-line for typical sets of average SNRs and the obtained optimal threshold and/or power ratio values are stored in a look-up table.

4 Simulation results

In all the simulations the noise components at the relays and destination are modeled as i.i.d.

CN(0, 1)random variables. For convenience, define

σ2_{= [σ}2

0,1 . . . σ0,2Kσ1,2K+1. . . σK2,K+1]. Figure 2 plots the average BERs at the destination for different channel conditions and different number of relays. Here the

threshold is simply chosen as_θth

r = 2to verify that our

BER analysis is valid for any threshold value. The transmitted powers are set to be the same for the source and relays. The figure shows that the analytical (shown in lines) and simulation (shown as marker sym-bols) results are identical, hence verifying our analysis in Section 3.

Next, Figure 3 compares the performance of the pro-posed scheme with that of PL scheme and the scheme in [26] in a two-relay network. The channel variances of all the transmission links in the network are set to be

s2

= [1.5 1.5 1.5 1.5]. The node energy constraints are

E0, max= 0.6Etotal,Ei, max= 0.3Etotal,i= 1, 2, 3. The

fig-ure shows that our proposed scheme with selection combining outperforms the PL scheme. This is expected since the continuous retransmission of relays in the PL scheme causes error propagation and hence limits its BER performance. Furthermore, it can also be seen that the relaying scheme proposed in this paper performs the same as the scheme in [26] under both cases of fixed and optimal power allocations. This is not a surprising observation either as it can be verified that in a two-relay network, whether selecting the best received signal or combining two received signals does not affect the

decision at the destination.d

Figure 4 shows the average BERs obtained by simula-tion for three different schemes in a three-relay

coop-erative network.e Heres2 = [0.5 1.0 2.0 1.0 1.5 2.0].

From the figure, both the optimal threshold scheme and jointly optimal threshold and power-allocation scheme achieve better BER performances compared to the PL scheme. The percentages of total power spent for node 0, 1, 2, and 3 are 52.47, 12.61, 15.59, and 19.33%, respectively when the average power per node is 20 dB. This optimum power allocation is reasonable intuitively satisfying since what it does is to allocate a big portion of the power to the source to reduce decoding errors at the relays. Then, more reliable relays are accordingly allocated more powers since the destination is expected to select the signal from the relay that forwards a cor-rect bit. Similar results are observed for other values of the total power.

Figure 5 presents performance improvement of the proposed scheme in a five-relay network when the

var-iances of Rayleigh fading channels are set to be s2 =

[3.5 2.5 0.1 1.5 0.4 3.5 2.5 0.1 1.5 0.4]. The node energy

constraints are set to be E0, max = 0.6Etotal, Ei, max =

0.3Etotal, i = 1, ..., 5. An SNR gain of about 3 dB is

observed at the BER level of 10-6 by the proposed

scheme with the optimal threshold value when com-pared to the PL scheme. The figure also shows that jointly optimizing the threshold and power-allocation scheme can be further beneficial in the proposed net-work. Specifically a further gain of 2 dB can be realized when compared to the case of solely optimizing the threshold value. The results presented in Figure 5 are also intuitively satisfying. Since the relays are geographi-cally distributed, the PL scheme suffers from more deci-sion errors made at the relays that are far from the source. Setting a proper threshold at the relays and/or re-allocating the power between the source and the relays is therefore beneficial in this situation.

It should be pointed out that the proposed scheme can actually save some power compared to the PL scheme (similar to the scheme with two thresholds pro-posed in [26]). This has not been incorporated in the BER plots in Figures 3, 4 and 5, where the BER curves

are plotted versus the average power assignedper node,

rather than the average powerconsumedper node. Such

a power saving is a direct consequence of the fact that a relay might be silent in the second phase. However, numerical results indicate that the power saving is sig-nificant only at low/medium SNR and without

power-allocation optimization.f

0 5 10 15 20 25 30 10í5

10í4 10í3 10í2 10í1 100

Average Power per Node (dB)

BER

PL

Opt. threshold [26]

Opt. threshold and poweríallocation [26] Opt. threshold

Opt. threshold and poweríallocation

Figure 3BERs of a two-relay network with different schemes whens2_{= [1.5 1.5 1.5 1.5].}

0 5 10 15 20 25 30

10í4

10í3

10í2

10í1

100

Average Power per Node (dB)

BER

K = 2, σ2 = [1.0 1.0 1.0 1.0]

K = 4, σ2 = [0.5 1.0 1.5 2.0 1.0 1.5 2.0 2.5]

0

5

10

15

20

25

30

10

10

10

10

10

10

10

10

Average Power per Node (dB)

BER

PL

Opt. threshold

Opt. threshold and power

í

allocation

Figure 4BERs of a three-relay network with different schemes whens2_{= [0.5 1.0 2.0 1.0 1.5 2.0].}

0 5 10 15 20 25 30

10í10 10í8 10í6 10í4 10í2 100

Average Power per Node (dB)

BER

PL

Opt. threshold

Opt. threshold and poweríallocation

source to reduce decoding error at the relays, and hence the relays are more likely to retransmit in the second phase.

Finally, it should be mentioned that, in general, the diversity order of the network depends on the chosen threshold value. Unfortunately, a theoretical analysis of the diversity order is not available. Nevertheless, the obtained BER expression is simple enough to plot and one can examine the diversity order by observing the BER curve. In fact, examining the BER curves indicates that the proposed scheme (with optimal threshold/ power allocation) achieves the full diversity order.

5 Conclusion

In this paper, we have obtained the average BER expres-sion for data transmisexpres-sion over a noncoherent

coopera-tive network with K+ 2 nodes. BFSK is employed at

both the source and relays to facilitate noncoherent communications. A single threshold is employed to select retransmitting relays. A relay retransmits the decoded signal to the destination if its decision variable is larger than a threshold. Otherwise, it remains silent. The destination chooses the received signal with the lar-gest decision variable to decode the transmitted infor-mation (i.e., selection combining). With the obtained closed-form BER expression, the optimal threshold or jointly optimal threshold and power allocation are cho-sen to minimize the average BER. Simulation results were presented to corroborate the analysis. Performance comparison reveals that the proposed scheme out-performs the conventional scheme with a similar complexity.

Endnotes

a

Considering orthogonal channels implies that one needs to trade multiplexing gain for error performance.

b_CN(0,σ2_{)denotes a circularly symmetric complex}

Gaussian random variable with variances2.cSpecifically,

we made use of the routine “fmincon”, which is

designed to find the minimum of a given constrained

nonlinear multivariable function. dWithout loss of

generality, assume that the first branch is selected to

decode the transmitted information, i.e.,θ1,3>θ2,3. The

decision is of the form: SC= |y1,3,0|2− |y1,3,1|2

0 ≷ 1

0.

With the scheme in [26], the decision is as

[26]=|y1,3,0|2− |y1,3,1|2+|y2,3,0|2− |y2,3,1|2 0 ≷ 1

0. One

can easily verify that both decisions give the same result

as follows: θ1,3>θ2,3⇔(|y1,3,0|2 - |y1,3,1|2+ |y2,3,0|2 - |

y2,3,1|2) (|y1,3,0|2 - |y1,3,1|2- |y2,3,0|2+ |y2,3,1|2)>0 ⇔Λ[26]

(2ΛSC - Λ[26]) >0. It means that if Λ[26] >0, then ΛSC

>0. Otherwise, ifΛ[26]<0, thenΛSC<0.eWe are aware

that the comparison between the PL scheme and jointly

optimal threshold and power-allocation scheme might be unfair. Since reference [25] does not provide an aver-age BER expression in a cooperative network with more than one relay, it is not possible to systematically obtain the optimal power allocation for the PL scheme. How-ever, we believe that our proposed scheme has a better BER performance than the PL scheme with/without

optimal power allocation. fTo keep Figures 3, 4 and 5

readable the BER curves taking into account power saving are not included.

Acknowledgements

This work was supported by an NSERC Discovery Grant.

Authors’contributions

HX proposed the new relaying protocol, carried out the simulations and participated in the draft of the manuscript. HH supervised the research and revised the manuscript. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 22 February 2011 Accepted: 21 September 2011 Published: 21 September 2011

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Cite this article as:Nguyen and Nguyen:Selection combining for noncoherent decode-and-forward relay networks.EURASIP Journal on Wireless Communications and Networking20112011:106.

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